Interpretations of Heyting's arithmetic—An analysis by means of a language with set symbols

Annals of Mathematical Logic 19 (I~JS()l 1-31 © North-Holland Publishing Comp: ny

INTERPRETATIONS OF HEYrING'S ARITIIMETIC--AN ANALYSIS BY M E A N S OF A L A N G U A G E WITH SET SYMBOLS* Martin STEIN Unit~ersily of MiJnster. MiinstJr~ Federal Republic of Germany

Received 17 October 197F, Well-known interpretatiors of Heyting's arithmetic of all finite types are the Diller-Nahm A-interpretation [I] and Kreisel's modified realizability, subsequently called mr-interpretation [4]. For both interpretations one can define hybrids ^ q resp. mq. in Section 4 a chain of interpretations---called M-interpretations--is defined (it was introduced in [6]}. filling the "'gap" between ,',-interpretation and mr-interpretation. In this paper it is shown that it is possible to prove in one stroke the soundness resp. characterization theorems for all interpretations of HA~<> (Heyting's arithmetic of all finite types with functionals for coding finite sequences). This is done by means of interpretations of systems which contain set-symbols. For these so-called M-interpretations, soundness-resp. characterization theorems can be proved simultaneously (Theorem 2.51. S,pecial translations of set symbols and of the t'ormula {Aw ~ W } A - - this means, special decisions about the size of the set W; see Sections 3 and 4 - - yield the corresponding results for all interpretations of HA,,< > mentk~ned above. The terminology of set theoretical language--we consider an extension of HA,o<, by an extensi'~ely weak fragment ~.~nly.which leads to a conservative extension of HA.,< >-- is of good use for studying realizing terms of different interpretations: if H A ~ >FA, A ~'~---::lt~Vw A~. and ~A~tltst. w l by soundne~,s theorem for M-interpretations. there exists a simple operalion which maps ~3~t to f.,~. the realizing term for modified realizability. For interpretations of Heyting's arithmetic--A-interpretation. M-interpretations and mr-interpretation--this leads to the following stability resull for existence theorems: if F::lxA and tA resp. t~ is the term computed by A-interpretation resp. M-interpretation. with FA[tM]. then --using extensional equality and o,-rule i'or equations--we can prove that tA = tM = tmr (Section 5). Remark. It is possible to extend these results to G6del's Dialectica interpretation. For this purpose, however, it would be necessary to construct a system HA,~ which is rather complicated.

Theo~Aes with set-symbols D e f i n i t i o n ! . 1 ( T h e t h e o r y T< >). (a) T y p e s of T<> a r e all l i n e a r t y p e s . (b) B a s i c t e r m s a r e (1) t h e c o n s t a , l t 0 o f t y p e o. (2~ l h e s u c c e s s o r f u n c t i o n N o f t y p e (o)o. (3) f o r e a c h n - t u p l e ~rt . . . . . ~r. o f t y p e s , w e h a v e f u n c t i o n a l s ( ) a n d j~. f o r * Sections I-4 of this paper arc based on results of the author's doctoral dissertation, written under supervision of Profe,,sor Dr. J. Diller. Miinster.

1

2

M. Su'm

coding a n d decoding, of a p p r o p r i a t e type, (4j functionals R~ . . . . . . Rk, for s i m u l t a n e o u s recursion. R~, h a s t h e type (o~h

• " " r,,'r0

• • - (or~

• • • ~'.%,)rl

" • • "r,,or~.

(5) variables of all types. (c) Inductive definition of t e r m s (1) every basic t e r m is a term. (2) if a is a t e r m of type (or),r a n d h is a t e r m of type 0r, so a i h ) is a t e r m of type ~'. (3) if a is a t e r m of type "r and x is a variahle of type ~r, then Axa is a t e r m of type (or)-r. T h e formal symbols of T< > are A, ---,. --. (.). (d) Inductive definition of f o r m u l a e (1) if a and b are t e r m s of equal type, t h e n a = b is a formula. (2) if A a n d B are formulae, then A/~ B. A ~ IJ are formulae. (e) A n a t t x a l d e d u c t i o n system for T<> T h e defining e q u a t i o n s for the basic terms and for the A - o p e r a t o r are the usual ones. For the n-place coding function ( ) which codes terms a'~,. . . . . a;~.,, we h a v e the axiom: /', (aT', . . . . a',,")=aT',

l~i~n.

T h e rules for /',E, AL ---',E, -..*I are the usual ones. T h e same holds for the induction rule a n d the rule for the equality sign. Definition 1.2 (The theory H A , , 0. Types, terms a n d prime f o r m u l a e of i-IA,.,,, are those of T< ~. T h e formal symbols are those of T¢ ~ plus V, :::1 T h e f o r m u l a e of HAo,<~ are defined inductively like those of T< ~, a d d i n g the following rule: If A is a formula a n d u a variable, then Vu A and ::lu A are f o r m u l a e . A x i o m s and rules of HA~< ~ are those of T< >. adding t h e rules r E , VL 3 E . ::lI. Definition 1.3 (E-types). ( 1 ) o is an e - t y p e . (2) If ¢r~. . . . . o', is a s e q u e n c e of ~-types. then {~r~. . . . . (3) If ort. or2 are ~-types, t h e n (or~)~r, is an c - t y p e .

tr,,} is an c - t y p e

T h e degree function g is defined on linear .',ypes only: g ( o ) : = 0,

g ( ( o ' ~ r ) : = max(g(~r)+ I, gir}),

W e m a k e the following a g r e e m e n t : x, 1 . 2 , 3 . . . .

In this section M is o n e of tile symbols

Definition 1.4 ( T e r m s of HA,,,M). (a~ Basic t e r m s arc (1) the c o n s t a n t 0 a n d t h e succe.,sor function N. (2) functionals ( ) a n d ]~ of a p p r o p r i a t e iiuear type for coding finite s e q u e n c e s of linear type a n d for decoding,

Imerpret~ttion~ of Heyting's arilhmetic

3

(3) functionals R ~ . . . . . R,,~ for s i m u l t a n e o u s recursion. (4) the symbol ~l~'~ of type {or} for all s e q u e n c e s o- of c-types. (5) the symbols ~c,,~ of type {o-} for linear tr. g f l r ) < M . (6} variables of all types. (b) Inductive definition of t e r m s of HA,,M T h e t e r m s are defined as in Definition I.l(c) adding the [ollowing rules: ( I ) if I ~, tt . . . . . t,, is a s e q u e n c e of terms of types or, . . . . . (r,, ~- m then {I, .....

~,,}

is a term of type {or}. FVI{II . . . . .

t,,}) = 0 FVlti}. i

BV({q . . . . .

I

t,,})= U BV(t~). i=|

(2) let W~ a n d IV: be t e r m s of type {or}. T h e n W, U W2 is a term of type {or}. (3} let IV, be t e r m s of type {¢r,}. Let w be a s e q u e n c e of variables of type ~r~. w~ FV(W~I. T h e n W,

U

is a term of HA,,,.~I of type {~r2}:

/

\

R e m a r k . {l~. . . . . t,} characterizes (I, . . . . . r,,~ So {tt. I : } # { t , . td.

the

"'singleton'" w h i c h i

c o n t a i n s the tuple

Definition 1.5 ( F o r m u l a e of HA,,,~). (I) if a a n d b are terms of equal type. then a ~- b is a formula. T h e r e m a i n i n g rules for construction of formulae of HA,o~.~ c o r r e s p o n d to those of H A ~ > a d d i n g the following rule: (2) Let or, . . . . . o',, ~-or be a s e q u e n c e of c - t y p e s a n d w " = - w , . . . . . w,, be a s e q u e n c e of lype ~r of variables. Let W ~'r be a t e r m of HA,,~.~ with w~ FV(WIm). Then {Awl .....

w , c W~"~)A ~ ( / ~ w c W ) A

M. Stein

4

is a formula of HA,,M. (read: " A holds for all elements of W"); FVt(A w ~ W ) A ) = F V ( W ) U ( F V ( A ) \ { w}). BV((A w ~ W ) A ) = B V ( W ) U B V ( A ) U{w}. For reason of simplicity we will sometimes write A w e W A (A w ~ W ) A .

instead of

Diller [2] restricts formulae I A w ~ W ) A to the case that A is quantilier-free. This restriction is not necessary itr o u r system, because the restriction rule of our system (see Definition 1.7) is easier to interpret than the corresponding rule in [2]. A x e a(B[x] ~ C[x]) ~ (A y ~ a B [ y ] --~ A z ~ a B[z]). Definition 1.6 Sequences are used as defined in [I, p. 52], T h e r e are only three new cases: (1) Let W ~ - W t . . . . . ;V,, be a sequence of terms of HAo, M of type tr~{,~,} . . . . . {o-,, L Let U be a term of type {~-}. Let u be a sequence of type 1- of variables with u 6 FV(U). Then

U w : - = u w, . . . . . U W,,. ucU

u~:U

ueU

(2) Let W -= Wt . . . . . {o-,, }. Then

W,, and V ~ V~ . . . . .

V, be sequences of terms of type

{oh} . . . . .

WU

V:-=- W~ U V~ . . . . .

W. U V,,.

Definition 1.7 A natural deduction system for HA,~M: "l'he axioms arid rules of HA,,,~ are those of HA~,< > adding the following rules for bounded quantification: 11 E ( A w e Wt U W21A U (A w~ W,)A !1 ,

E

(AveU.,.wlA,dv]

.

I1 A c (_A u ~ U)(A w ~ W)(A u ~ U ) A w~ FV(U), (A u E UI(A w c W t A II

{}E(AW~lt,

II

. . . . . t,,})A

•% D , . . . . .

t,,[

;k(Awc/~"bB B~[t]

if o- is small with respect to M (see Definition 1.8).

lnterl)retations of Heyting'.s arithmetic

5

Restriction rule. Let A~ • • • A .

II B be a deduction of H A . , ~ such that all free assumptions of 1! different from A~ . . . . . A,, do not contain w free. Let -vi

(i = I . . . . . ~1)

A w e WA~ be deductions of HA,,,M. Then v

...

(w~W) Aw~WA,

v

Aw~WA,~

A~

A,, Ii B

Aw~WB is a deduction of HA.,M. Remark 1. Some!'mes we shall write A~ . . . . . A . ~ B for: "There is a deduction of B in the system ~ (~ any of the systems T~M, HARM. HA,~(>. T< >. TA~ >) with free assumptions A~ . . . . . A,,.'" Remark 2. Let us consider the following special case of the restriction rule: If A

II B is a deduction, w not free in any assumption difl'erent from A. then

(w~ W) A w ~ W A . A II B

wE W B is a deduction. The underlying idea is to imitate for arbitrary sets the following method of constructing restricted deductions which should be correct in any system which contains finite sets {It . . . . . I,,}; {tl . . . . . t.} should be understood here (and otlly here!) as the finite set containing the terms t~. . . . . t.:

M. Stein

6

If A II

B is a deduction, then (Awe{t~ . . . . . t,})A

( A w eTtt . . . . . t,})A

A[t,] !1 [h]

A[t,,] [t,,]

It

""

n[t,] {Aw et(~ . . . . . t,,})B should be a correct deduction too. R e m a r k 3. It would be natural for a system containing the symbol e to accept t'" e W ~''1 as prime-formula. T h e n one could introduce the following rules for A e :

[we w] il AE teWAweWA Aw[t]

AI

A Awe WA

An application of A I is correct only if w is not free in any free assumption of [1 different from w e W. By means of AE, A I the rule A e would be derivable, and so would be the restriction rule:

[we W]Awe WB~ ...[we W]Awe WB,, B~ B. 11 C Awe WC In our system HA,,r,~ the restriction rule is marked with ( w ~ W) because the above derivation contains the b o u n d prime-formula w e W. T h e reason for not accepting t" e W ~'~ as prime-formulae of HArM is that t e W cannot be translated to a prime-formula of TA~ > resp. "/"<>M (Definition 3.0. and Definition 4.2.). On the other hand, the restriction theorem is strong enough for proving Theorem 2.5. Definition 1.8 Let M e { ~ } U I q . Then we define for E-types tr: o- is small with respect to M:C~(tr is a linear type and g ( t r ) < M ) or M ~ : ; cr is big with respect to M :¢~ ~r is "~ot small with respect to M.

Interprelations of Heyting's arithmetic

7

Definition 1.9 (M-formulae). M 4 o r m u l a e are formulae of HAo,M which satisfy the following conditions: They contain no existential quantifiers, and il Vx" A is par: of a M-formula B, then ~r is small with respect to M. Definition 1.10 (T,~). Let M e {~} U N. The formulae of T~.,a are all M-formulae. The rules are those of H A j ~ , but the rules VI and VE are restricted to variables which have a type small with respect to M ; the rules 3 I and 3 E are cancelled.

Theorem 1.11. The inference rule

[Au ~ WA[x, ul] Ila lid A[Nx, z]

A[0. z]

Aft. z] is admissible in T~M if 11.,. 111 are dcductions in T~_M. and if A u ~ W A [ x , u] is the only free assumption of lit, and if 11. is closed. Proof. Let ~V:=-=R (AxAUAx.Az

U,. W ~ [ z . , x o ~ N x ] ) (AxoAz{z})

(z~ is a new variable). Then the following holds:

~--WOtz={z}~---i£VNrtz=

~

W~,[z,,t'-Nr].

z : Wrtz

The wanted deduction is as shown in Tableau 1. Tableau 1

Nx:f=O x:t=O~(Az,~V(t:xlzz)A[x.z~] Nx~t=O { A , z ~ ~ V ( t : x ) t z ) A [ x . z t] t ~ x = N { t - Nx) f A zl ~ ~V(N(! : Nx))rz)A[x. zl] {A z l ~ U : ~ . ,&l, ~ , , ~ W ~ . [ z p ! : N ( t : N x ) ] ) A [ x . : 1 ] Nx :t = x-:t=0

¢A z~ e "~/(t:Nx)tz)(A u ~ W~.[zp t :N(t : N x l ] ) A [ x . u] t : N ( t = N x ) - x (z:e~V(f=Nx)tz) {AzlcW(t=Nxltz){AucW~[zl])A[x.u ] C (A u e W~[z.]A[x. u] It.

II ,.:[z,] A [ N x , z ~]

A[o. z]

{A :~ e f.Vuz)A[O, z~] {A z~ c ~!( ~-{}~lzlA[0,z~]

{A z l ~ ~Vtt = Nxltz)A[Nx, zl]

0 = I = 0 ~ ( A z] e7 ~'(t = O}lzA [0, z t] N x = I = 0 - ~ {A z I ~ 17V(I = Nx)tz i A [ N x , z t ]

t = t =O

I=l={}-...*(A z l E QC{t~t)Iz}A[t, zl]

{A z~ ~ WOtz)AI t. z~]

IA z, ~_{z})A[t. z,] nit, :1

8

M. Stein

2. Interpretations of H A r m Definition 2.1. Let t =--t~. . . . . t,, be a sequence of terms with type o7 . . . . . tr,., Then we define: t~ is the subsequence of those terms of t whose types are big with respect to M, t ~ is the subsequence of those terms of t whose types are small with respect to M. If there can not be confusion about what M is meant we write i" instead of t u. I instead of t~ Definition 2.2 (MP-interpretations), Let P be a function on the formulae of H A ~ f which satisfies the condition that F V ( P ( A ) ) c _ F V ( A ) . Let M ~ I U { ~ : } . Then the chain of MP-interpretations is defined as follows:

(1)

(tl=tz)M~'~tl=t2=--(tl=t2)~.

Let A ~P =-3 v '~ V w "~A~e. Then we define:

with disjoint sequences v, w, y. z.

B Me -':--3 y ° ~ z " BMj,

(2)

(A A B )~e _=::Iv y V wz ( A

(3)

( A - o B)MP-~3W~J~'Iy"~ Vf~z~(A-'~ B)Mp,

A

B )~t,,

( A ~', B )Me ~ AMp/', BMx,.

(A --~ P)Mt, ~ V~)(A w" ~ W v z AMI,[u, W]A P ( A )--* BMp [ Yc. z ]). (4t

(Au~

WI~" A ) M P ~ 3 V I ~ ' J " V w ' ( A u ~

(A u ~ E w ~ A)Mt, ~ A u ~

Wfm A)Mt,,

(weFV{WD

W ~m AMe[ Vu, w],

(5)

(Vx~ A ) M e = - 3 V ~ n ~ V w x ( V x ~ A ) ~ e ,

(6)

(3x ~ A )~e =-3xv V w ( ~ x ~ A )~p,

i

(Vx~ A ) ~ e = - V ~ A ~ e [ Vx. w. xt. (3x ~ A )Me =- P ( A )/', AMp.

Diller [2] gives a good description of the underlying idea eft tlle inlerpretatlon of A - - . B : "Given a deduction II of a prenex formula 3y V z B from an assumption 3 v Vw A, we should be able to construct from IL for any given v. an object y such that. for any z, every path in H leading upwards to an occurrence of the assumption 3v Vw A gives us an object w which in this path is the reason for A to imply B ; as different paths in II may produce different objects w, the deduction II as a whole only gives us a set W of objects (reasons} w such that Ii contains i proof of B from assumptions A for all w c~ W. In short, from II we obtain a proef of Vv :ly Vz 3set W ( A w ~ W A ~ B).'"

(p. 149.)

We will show in Sections 3, 4 that different interpretations of sets W lead to A-interpretation, modified realizability and the M-interpretations which lie between A-interpretation and modified realizability.

Interpretations of Heyting 's arithmetic

9

We used "~quantifiers of limited type" in Definitions 2.2(3), 2.2(5) instead of the possibility (3)

(A ~ B) ~''b ---3W ''c*~ Y"" V v z " ( A ~ B)~Vp, (A ~ B)~7, - A

(5)

w ~ ~ W v z A ~ [ u , w ] ^ P ( A ) --, B ~ [ Y v , z],

(Yx~ A)MW'--=3V'ti'Vwx(Vx~ A)~'k, (Vx ~ A )gTb =- A~F[ Vx. w. x]

to obtain identity of the sequence of M-interpretations (Section 4) with the sequence of n-interpretations [6], and identity of ~-interpretation resp. ~ interpretation with modified realizability. Definition 2.3. (1) In case that P(A) -= 0 = 0 for all formulae A the corresponding MP-interretation is called M-interpretation. (The prime-formula 0 = 0 may be omitted without loss.) Instead of A Me we write A ~, instead c,f A~p we write AM. (2) In case that P ( A ) z ~ A for all formulae we call the corresponding M P interpretation Mq-interpretation. Instead of AMP we write A ~". instead of AMp we write AMq. Remark. In case that M - = ~ one easily sees that for all A it holds that if A ' ~ = 3 v VwA~,~. then w is the empty sequence. In fact, the zc-interpretation is just the mr-interpretation for H A ..... Equally, the ocq-interpretation is the mqinterpretation for H A .... In case that M = 0 the formula A ~ contains no universal quantifier. The 0-interpretation is the HAlo-analogy to the Diller-Nahm A-interpretation for HA~¢ ~. Lemmz 2.4. Let A be a formula of HARM. Then (1) AM is a M-formula. (2) Let A be a M-formula. Then T~FA

"~ ~'-~AM, ~--~A;

A~=-A~a~A.

Theorem 2.5 (soundness theorem). Let P be a function on HArM as defined in Detinitioa 2.2 with the following properties (M ~ F~ t3 {7:}): (I) A~ . . . . . A,,FllA~.,, H ~ P ( A O . . . . . P(A,,)FHA.,P(BI; (2) HA~.~f ~---P(A)A P ( A ~ B) --, P(B); (3) P(A~[t]) ~ e ( A ) d t ] : (4) P ( A w e W A ) = - A w ~ W P ( A ) . Let il be a natural deduction of HA~.~ with free variables A , . . . . . A . and endfonnu~a B. Let B t c a ' = - 3 y V z B ; A~r'=-3v, Vw~.A,. Then there exist tenns W t . . . . . W,,. ~ and a deduction H t'w such that li Me has the formulae Z w, ~ W~ fit~ and P(A,) as free variables and the endformula B[~, z].

10

M. Stein

For W~ a n d ~ it holds that FV(W~)~ 0 FV(A~)UFV(B)U{vt

.....

v,,, z},

i=l

FV(f) 0 FV(A~) U FV(B)U{v~ . . . . v,,}. Proof. The theorem is proved by induction on the deduction of B. We only consider the cases that the last inference of II is an application of ---->I, ---,E and the restriction rule. The last interesting case--induction--is interpreted by means of Theorem l. 11. (I) The last inference is a ---,-introduction. AI, A2 !1 B A~---~ B ~ • Let A M, P ----~v, vw,,~,, B ~ ' - = 3 y Vz/~. By induction hypothesis there are terms W1, W > ~ and a deduction I! My such that

P(At)(A wl~ Wi),S,i ,

,

(A u,z,~w.~).~2 P(A~) MP

II

Lct tr be the type of it. Then the wanted deduction is as shown in Tablcau 2. (A w~c U r r ,-.~,.',w_,)A: (A f-'tc~o~"t)(A w~~ W.)J~ A w,e w, Xt P(A~) A w , ~ W j , 2 P(A:) 11

hi9, zl /]l(,~t'~ 9)t', : } P(Ao-~ [~[(At'~9~el, zl A w~c-W 1Ait, P(At)~fft[(Al:l~)l:t.Z ] (AwL c (AvtzW~)e~z )fi, I ^ P(A 0 ~ /~[(Avl~)t7, z] V~'I((Aw I ~ (At:jzW O~:lz)fi~~,~ P(A]) ~ /~[(At:l~,)t7, z]

'fubleau 2. (2) The last inference is a --,-elimination, A S C

A 11 C-~B B

Interpretations o[ Heyting's arithmetic

lI

Let A ~P -~]v V w , ~ , B MP ~ 3 y V z / ~ , C ~ e ~-3131VW t ~'.

By the induction hypothesis there are terms Wt, W:, W~, 13. ~' and deductions v ~ P i i M~" such that the following holds: P I A ) A w e W~ A .,v--T- ~,t,

OIf~. w,]

P(A)Awe

W~

MP

lI

V~,(tA w, ~ W~v~z)O[~,. W,]AV(C) --./~['2~,. z])

By conditkm on P there exists a deduction ..1 P(C) "

The wanted deduction is as shown in Tableau 3. ~Aw~ O~,,. ~.,~= W~u W~,,[~l) fit Awl: U ~ r wy: Wl fit (w~ ~- W ~ f : z ) ( A wI~_ w - , f z l l A w 6 wt)fit t~df,

--

C[O. w~]

P(A O,¢X w ~ W=~L[~]) A - --t.~P -

.1 (A wt~ W~fz~('[~.w~] t'(C~ IA w~E:W,ez)C[e. wOA Ptc) ~

B[Yf.

z]

Tableau 3 (3) The last inference is the restriction rule (we only will consider the special case mentioned in Remark 2): A II~ B

( w e V,')A w e W A A II B Aw~WB

Let A ~ ' = 3 x V u f i L .

B~'-~3yVz/~

(u¢FV(Wt).

Then

IA w ~ WAW"

~3x VulA w ~

w,S,.[Xw]).

(A w ~ W B ) ~ ' P = - 3 Y V z ( A w ~ W B ~ [ Y w ] ) . B~¢ induction hypothesis there are terms U. ~ and a deduction II 'va" such that ( A u ~ U)fi,, P(A)

~[y, z]

M. Stein

12

The wanted deduction is as shown in Tableau 4. (A u ~ U ~ w U~[Xwl(Aw e w)A,[xw] (A w e W)(A u e U~[Xw])(Aw e W;A,[Xw]

P(Aw ~ WA) (w~W)(Aw~W)U\ueU~[Xw])A,[Xwl

(weW)AweWP(A)

(A u E u,[Xwl)AJXwl, i'~A) lr "',[x~l B[(Xwg,[Xw])w.z]

(A w e W)~[O,wg~[Xw])w, z] Tableau 4

Corollary. In case that the MP-interpretation is the M-interpretation. II ~P is a deduction of T~M. So HArM is col~servative over T~M. Definition 2.6 Let M ~ N U{~}. Le~ IPM be the scheme (VwA---->3xB)---->3r(VwA ~ B)

for M - f o m m l a e A ; x¢_FV(A),

Let MtM~ be the scheme ( V w A ~ B)-->r:IW(A ~,~ W V # A ---, BI

In case that N-~ A read: ~ e

for M-formulae A, B.

W V ~ A ~ B =-V w A --, B.

Let ACM be the scheme V w 3x A - , 3 X V w A~[Xw] for formulae A of HA,.,,~

Remark. M(~ is derivable Theorem 2.7. The rules corresponding to the schemes IP M. M(M), AC~a are admissible over HARM. Proot. The theorem is proved by means of the s,~uudness theorem and Lemma 2.4. Theorem 2.8. (Characterization of M-interpretations). The theories HA,,,M+{A ~-'>AM: A formula of HA,,,M} and

HA,,M + IPM + M(r,~,+ ACM are equivalent.

This theorem may be proved by the methods shown in [1, pp. 55-56]. Remark. The schemata required to characterize M-interpretations have the following property: From 0 to 7: the IP-schemata get stronger and stronger, whereas Mo is the strongest Markov-scheme, and M(..) is derivable

Interpretations o[ Heyting'.s ari hmeric

13

Lemma 2.9. Let I~M(~'CM) be the scheme I P ~ t A C ~ ) restricted to formulae of HA.,< ~. Let I(4o4~ be the scheme MtM~ restricted to [ormulae V w A ~ B of HA~< 5. Then HAo,M+{A'~.AM: A formula o[ HA,,< >}~---D

if D is an instance of IPM, ACM or MtM~. The simple proof is left to the reader.

3. Applications to the DUler-Nahm A-interpretation In this section we want to apply the results of Section 2 to the Diller-Nahm A-interpretation which i s - - g e n e r a l i z e d by special functions P on the set of formulae of HA,,,< >-- defined in 3.1. Because of this intended application we will m a p terms and formulae of HA,~, in such way to terms and formulae of HA~<> (by mappings Z resp. A ) that for formulae A of HA,< > holds: A (A")=- A " . Because of this intention we will interpret A w " ~ w ~ A (in case that A is a formula of HA,,~. or--- o'~ . . . . . ~r,, is a sequence of linear types) as a formula

A x < x .aw[~/x].

~ ----~ ? ' " , . . . . .

~'I: ''''o,

a n d this means that we intend the following notion of set: sets W are interpreted to have finite size. that means they represent a finite sequence a~ . . . . . ax (X not necessarily closed), Consequently, S (W~"b has to be the sequence of terms X. Ib'~. . . . . l,i,~. These reflections will be considered in Definitions 3.3, 3.4 and 3.7.

Definition 3.0 (The theory T~< >). Types. terms and prime formulae of T,<> are those of ~1"~ (Definition 1.1.). T h e formal symbols are those of T<5 p l u s / ~ <. The formulae of T.<> are defined inductively like those of T~5, adding the following rule: If A is a formula of To<~. X a term of type o not containing x free. then A x < X A is a formula of TA< ~. Axioms and rules are those of "I'~> adding rules for bounded quantification:

[x
Ax
A AIAx
A n application of A I is correct onlv if x is not free in any free assumption of II different from x < t.

14

M. Stein

Definition 3.1 (A P-interpretations). Let P be a function on the formulae of HA°,<5 which satisfies the condition F V ( P ( A ) ) c _ F V ( A ) , T h e n the / , P interpretation on formulae of HA,o< > is defined as follows:

(l)

(ll=12)^P~-tl=12-~(l~=12)^p,

Let A ~P ~-::]u Vw A,,po B ^~" ~ t y Vz B~p with disjoinl sequences u, w, y. z, T h e n we define: (2)

(AAB)"P=-3vyVwz(AABLp, (A A B)^p ~ A ~ p A B~,p,

(3)

( A - * B)'P=-3XWYVt'z(A--~ BLp (X, W. YC.FV(A,,p)UFV(B~p)),

f A X < X l~ZA^p[ W l ) z x ] (A

t (4)

A

P(A) ~ B^p[Yv. z]

if w ~ A (x a new variable)

B)~p A~p

-.-> ~J A p [ Y t J ,

z]

if w-~A.

(A x
(5)

(VX A) ^P =-3V Vw(Vx A)~p,

(Vx A L p ~ A,p[ Vx], (6)

(3x A FP ~::lvx Vw(3x A t,,p, ( B x A L p ~ P ( A ) A A~p.

Definition 3.2. (1) In case that P(A)=-0 = I) for all formulae A the correspondi~lg /',P-interpretation is the D i l l e r - N a h m A-interpretation, if we omit the primeformula ()=(t. We write A ~ instead of A "v. A~ instead of A,p. (2) In case that P ( A ) ~ A for all formulae A we call the corresponding A P-interpretation A q-interpretation. Instead of A 'q' we write A TM. instead of A~r, we write A.... Definition 3.3. A mapping ~ of z-types to linear types (o) : ~ o,

(~r~ . . . . . ~r,,}) :~: o. ( o ) 2 (,r,) . . . . . ( o ) ~ (,r.). Remark. (o-t . . . . . ~r.)-r~. . . . . ~-,, is defined as (7; I " • • ( r , , T t . . . . .

or 1 " " •

Cr.y..

Definition 3.4. A mapping 7, of terms of HA,.~. to terms of HA,.,< >.

Interpretations o[ Heyting's a, ithmetie

15

The mapping is defined inductively on the definition of terms of HA,,,. (a) Basic terms (1)

S(O ~', ...... "o~):~(t, AxO~('',). . . . . Ax()~"',a,

where 0 '~'',~ is the sequence of 0-functk)nals of type ~ (o-~). (2) Let w" be a variable of type or. X(og-=-¢r~ . . . . . rr,,. Then (3) Z ( N ) : ~ N ; Z ( 0 ) : ~ 0 . (4) The other basic terms are mapped similar to the mapping of R~:

(b) Interpretation of terms of HA.o (1) Basic terms are interpreted as above. (2) Z(a(b)):-=- Z(a)(Z(b)). (3) Z(AxO:=--AZ(x)Z(t). (4) S({q . . . . . t,,}):=~l, A x Z ( q ) . . . . . Ax/~(t,,). x i=l,....n. (5) Let 7,(W["r)~X,, I~'~ (X~ of type o). Then

of

type

o, xc~FV(t~),

-Z (W, U W,):~- X~ + X:." fV. fV :=- Ax( U x ( N x ":- X~)). The sequence U is defined bymea~.s of fur ctionals for simultaneous recursion such that the following holds: UxO = ~Vtx. U x N y = W2(x ~ X 0 . ~ ' is constructed in such way thaL if the s~quence a . . . . . . . ax, ~ represents

Z(W~).

b,. . . . . . bx: ~ represents

~(W,).

then a,,. . . . . a×, i.

ax,. . . . . at×,~x: i~ with a(x,+i):=bj represents

Z (W~ U W2):

if x < X j . then Vv'x= W~x, if X t < ~ x < X 2 , then 17¢'x= W2x. (6) t.ct Z ( W I " . b : ~ X , . W,. Let xt be a new variable. Then :~ (w,Y,v, W e ) :-=-
Ax(~V2~.~.,ffV,hxlj2x).

The term T:=-
A X
16

M. Stein

Lemma 3.5. (1) If t ~ is a term of HA,.. of type ~r, then -~ (t) is a term of HAo,< > of type ~ (cr). (2) For terms t of HA.,<> it holds that -A (t)w. t. (3) For all terms t of H A ~ it holds that FV(-A (t))= Z (FV(t)).

(4)

S (t,~[t,]) -= Z (h)~c~Z (t2).

Definition 3.6. A mapping A of formulae of HA,,, to formulae of HA.,~ >. (1) Let Z(t")=-tt . . . . . t,,, X(r")=-rt . . . . . r,,. Then A(t,, = r , , ) : ~ t t = r t / x . . . At =r,.

(2) ^(A~-~B):=-A(A)-~,A(B) ^~x A):=--~,(x)^(A). (3) Let S ( W ) - = X , V¢ Let x be new for X. Then ^ ( A x e W A ) : =- A x < x A (A L..~,,~[1,Vx].

Lemma 3.'/. (1) I r A is a formula of HAlo, then A ( A ) is a formula of HA~<>. (2) I r A is a formula of T~o. then A ( A ) is a formula of T.<>. (3) For formulae A of HA~< > it holds that /~ (A) =----A.

(4) /,,(a,~[t])--- A(A),~,,,,[Z(t)]. (5) Let b be a function as in Definition 2.2 such ~hat P = b [ {A : A formula of HA~< >} is a function as in Definition 3.1. Then for all formulae of HA.,~ ~ it holds that ^(A°O)~A ^r',

A(A,,o)~A~,.

The proof is left to the reader. Theorem 3.8. (a) Let 1; be a deduction of HA,.,, with free variables Bt . . . . . B . and endformula A. The~; there exists a deduction II ^ of HA,~< ~ with free variables A(B0 . . . . . A(B.) and endformula A(A). (b) If H is a deduction in T~o, the deduction I F can be constructed in "/'..~>. Proof. We only consider statement (a). (One easily sees that the deduction I1 ~ constructed is in T~<> if !1 is in T~o.) Theorem 3.8(a) is proved by induction on the dcduction of A. The only interesting cases are those of UE, U~. E (1) Let fi. : ~ A(A), X. I~¢,:~ 7~(W,). By induction hypothesis there exists a deduction /) of the formula

A(A we W, U W 2 A ) ~ A x < X , + X~fq~vx]. ~/ as defined in Definition 3.4(b(5))

Interprelations o[ Heyling ' s arithmelic

17

The wanted deduction for A x < x 2 fi.[t~Czx] is as shown in Tableau 5 [x < x2]

/7

x*X,
Ax
~,[U(x + Xj)Nx] U(x + X,)Nx = fV2(x+X0 ~ X~ A[ g'~(x + X0 ~ X~] (x + X0 ~ Xl = x

'~[g'M

A x < X~ ~[¢¢~x] Tableau 5

The deduction of A x < X~ .A[Wix] is a little bit simpler. (2) t.ct ^ ( A ) : ~ / / . , /~(Wi):~X,. lg',. S(v,,0:=-g't, ~(W2):'~@2"Then A (A w : e

U

W2 A ) ~ A x <

wlEWl

TA.~.[Xx(ffV2.,D/Cj,xlhx)x].

the term T is defined by:

T : ~
/~ ( A w 2 c

U

W2A).

The wanted deduction is as shown in Tableau 6. [x, < x,l[x, < X.~[Cc,x,ll x~
{x~
A(Aw~cU~+ow, W~A)

III A x < TA%[Ax(fV, c~,[~Vlitx]i2x)x]

A~,[~xtg'..,[@,i~xli_.x)(x,. x~)l A~.[gV2~,[@,i,(x,. x.)li..(x,. x2)l A.~;[¢¢..,[ g',x,lx..]

A x. < x,~,[g/,xjA~,[Cc..,[fv,xdx.l Ax, < x, A x:< x._.,[fv,x,la,~[fv..,[fv,x,lx._]=- ^ (Aw, ~ W,Aw_.~ W2g~. Tableau 6

Theorem 3.9 (soundness theorem for r, P-interpretations). Let P be a function as

defined in Definition 3.1 which satislies the following conditions: (I) A, . . . . . A,,~-ItA.,,A ::~ P(AO . . . . . P(A,,)~"~A~,~P(A), (2) H A ~ >~-- P ( A ) ^ P ( A "-~ B) ~ P(B), (3) P(A~[t]) =-e(A)~[t]. Let there exist a function P for HA.., satisfying the conditions of 2.2 and 2.5 such that P t {A: A formula of HA~¢ )} = P. Let Ai . . . . . A . ~'~aA~,, B, B ^e =__=lyVz B,

Ig

M , Slein

A ~ e - ~ v , ~w~Ai. Then there 1re terms X~ . . . . . X.. W~ . . . . . W.. 9 such thai P(A,). ~ A ~ , , < > : , ~ , [ W , x , ] ~ - - - a r f i=1

i=1

HA"I

>

~]

"

and

FV(X~)UFV(W~)~ FV(B)t3 ~J FVIA,)L:{vl . . . . . v,,. z}. FV(9)c- FV(B)U 5 FV(A,)U{v~ . . . . . v.}. i=l

If :~ P is the A -interpretation, the deduction of B[f;, z] can be found in T~< >.

Proof. AI . . . . . A.~

B ~

A~ . . . . . A , , ~ - - ' B

This implies (see Theorem 2.5) that there arc terms ~/~ . . . . . I,~:,,, 0 such that

FVI~)~_FVtBtU U FVIA,)U{e,..... e,,..% FV(9)~ FV(B) U

i:l

i:

I

FV(A, JU{~ . . . . . 5.,},

HA

....

"

implying (cf. Theorem 3.~) that there are terms X~ . . . . . X.. Wj . . . . . W,, -y such that

A (P(A,)), ~ A x,
i-I

This is the statement of the theorem if we use Lemmas 3.5 and 3.7 and the fact that A(P(AI))~- A(P(AI))=-P(Ai). Corollary: Soundness Theorenr~,~ ]br A- and A q-inlerpretaliotj, Theorem 3.10, (Characterization of the /,,-imerprelalion). Lel :vL be the scheme (Vw A ~ B)---~BX, W ( A x ,z X A [ W x ] - - ~ B) for formulae A, B of T:,~>. Lel IP~ be the scheme (Vw A -"> 3x B)-'~ ~ x ( V w A --> B)

Interpretations of Heytitlg's ~rithmetic

19

for formulae A of T~< ~. B of HA,,,< >. Let A C be the scheme 'fix 3y A --~ 3 Y Vx A[Yx] for formulae A of HA,,,< >. Then ttle theories HA,,,< >+ {A ~--~A ~}

and HA,,,<>+ IP:,+ AC + M~ ore equir,alent. Proof. Let /9 be an instance of the schemata/Qo. A ' ~ , ~ 2.9. Then by Lemma 2.9. It holds that

as defined in Lemma

{A<-->A°: A formula of H A , >}~------/~. HAo,.

By Theorem 3.8. one easily sees that { A (A "~ A"): A formula of HA~,< >}~HA,,.A (/5). This means by Lemma 3.7. {A " ' ~ A ' : A formula of

HA,,< )}IHA,.., A (/~).

Now if / ) is an instance of one of the schemata Mo, AC,., IP,,, :, (/)) is the corresponding instance of the schemata MA, IP~. AC. Because for any instance D of the schemata M:,, IP~, AC there is a corresponding i n s t a n c e / ) of the schemata IP,'~./Qo, AC"~,such that /~ (/~)-= D, it is proved that

HA,o<>+{A~->A^: A formula of HA~<>}~---D for any instance D of the schemata IP^. M^, AC. To prove HA~,<>+M^+IP^+AC---A ~ A ^ we use Theorem 2.8.: HA,,,., + MI~ + I P . + A C , ~ A <"~A". Now let A be a formula of HA~< >. Then by Lemma 3.7 and Theorem 3.8 we get HA,~< >+ A (Mo) + A (IPo) + A (ACo) ~-- A <"-~A " . Because all formulae of /~(Mo) resp. A(IPo) resp. A(AC.) are instances of M, resp. IP:, resp. AC. we get HA,,~ >+M~ +IP^ +AC~--A~-->A ~.

M. Stein

20

4. M-interpretations and mr-interpretation In this section a new sequence of interpretations of HA,o< > is constructed, called MP-interpretations. Subsequently we will apply the results of Section 2 to these MP-interpretations (for special instances of P), Simultaneously, we will receive the usual results for Kreisel's mr-interpretation as Soundness Theorem and Characterization Theorem, We make the following agreement: in this section, M is any of the symbols i. 2, 3 . . . . or oc It holds that I < ~ , 2<~-, 3 < ~ . . . . and it does not hold ~ < ~ . W e define: ~/:={M-I

ifM~, if M =-~:.

Definition 4.1 (M-formulae). A formula of H A ~ >is a M - f o r m u l a if the following conditions hold: (1) A does not contain any 3-quantifiers. (2) If A contains V x ' B as subformula, then g ( r ) < M.

Remark. All 3-free formulae are ~-formulae. Definition 4.2 (The theories T~M). Formulae of T , , ~ are all M-formulae. The rules are those of T~> including the following two rules: II A VIM - Vx~A

if x is not free in any free variable of II and g ( r ) < M , II Vx'A VEM -----

A~[t]

if g(q') < M.

Definition 4.3. Let t ~ t ' ( , . .

,. I',;r be a sequence of terms. Tt~cn tM is the subsequence of t consisting of all t;', with g(o-,)~>-M, t M is the subsequence of t consisting of all t'[, with g(tr, l < M. Again we write ? for t M, L for tM. if there are no possible misunderstandings.

Definitior: 4.4 (MP-interpretations). Let P be a mapping of formulae of HA~,~ >to formulae of HA,o<> such that holds: FV(P(A)) c_ FV(A).

lnrerpretafioas of Heyting" ~ arith metic

21

T h e n the sequence of MP-interpretations is defined as follows: (I)

(a=b)Mm~a=b~(a=bjr~.

Let A~r'=-3u" VW'AMt,; y, z. T h e n we define: (2)

BMV=--3y" Vz"BMe with disjoint sequences v, w,

(AAB)Me-~3vyVwz(A/~B)rm,. (A A B M v -= A ~ A B~n,.

(3)

(Ax
(4)

(A --~ B)MI'~3W YV~z(A --o B)~a,, (A---~B)

=[V~'(Vu~;IA~,[v, Wvzu]AP(A)~'~BMv[Yv, z] Me- ), V~(A~,/x P(A ) --~ B~a,[Yr, z])

(5)

(Vu'B)Ma'-=3YVzu(Vu B)Mp,

(6)

(3u B)MV-~quy V z ( 3 u B)MV,

if w ~ A , if w - A.

( g u B)MV~V~ B~u,[Yu. z].

(3U B)MV~ BMv/x P(B).

Definition 4.5. ( 1 ) In case that P(A) --- 0 = 0 for all formulae A the corresponding M P - i n t e r p r e t a t i o n is called M-interpretation. (The prime-formula 0 = 0 may be omitted without loss.) Instead of A F~n"we write A M. instead of AMp we write AM. (2) In case that P(A)~-A for all formulae A the corresponding M P interpretation is called Mq-interpretation. Instead of A ~ (AMp) we write A Mq

(AMq). R e m a r k . T h e ~ - (resp. ~q-) interpretation is identical with the mri n t e r p r e t a t i o n - - K l e e n e ' s modified r e a l i z a b i l i t y - - ( r e s p , mq-interpretation) for HA,,, >. To apply the results of Section 2 to these MP-interpretations we have to define a mapping M of formulae of HA~,M to formulae of HAw<). We want to map formulae of shape /~ w" ~ W I'~ A in the following way (for reason of simplicity we here will only consider the case that w '~ is a single variable of linear type, and A is a formula of HA,~ )): ( l i If g ( t r ) < M we want to choose the "'thickest'" notion of set:

M(A w" ~ W~"~A)=-Vw"A. So it will be practical to m a p {tr} and W ~"~ in this case to the empty sequence, (2) If g(¢~)~>M we want to e n u m e r a t e the set W I'~ by a sequence of type /f/. This means that we have to construct a term I~' of type (/~/)o, such that M ( A w ~ ~ W~"JA)=-VxMAw[¢CxM].

22

M. Stein If M=--w-. the second case will not appear.

Definition 4.6 (Convention for the empty sequence ,4). (A)r:~r

(A)A : ~ A

(r)A : ~ A

A(a):~:l

a
A(A):+~A

a.lt:~t

AvA:~A

v AA :-~-A

~xA:~A

Ax
A/~A:=-A A---~A:~A

A-+A:~A

A--*A:~A

AAA:~A

A=A:~A

Definition 4.7. A m a p p i n g / ~

AAA:~A

of ~-types to linear types.

(1) /Vt(o) :-= o. (2) ~ ( ( c r ) r ) :~- (~(cr))/C/(r), (3) Let cr~-oq . . . . . o'.. ~l(cr+)~=cr+,, . . . . . ( N e only present the non-trivial cases.): (a) Basic terms (I)

/9/(R~):~JA. if ~ffr) :=-A, LR ~w,, otherwise;

(2)

/~(9){"' ....... "+):--= a x ~ O ,, . . . . . hx~aO .... if /V/(I+r+. . . . . m,})~rL . . . . . r,,.

(3)

/~(&:"}) :~- A

(:)

JVltw"):-= ~A

A

if /V/l{ct, . . . . . ,r,,}) ~ .I.

(g(o-) < M),

( w'~',

if /V/(o-I--=A, ....

w~,"

if /~/(cr)-=- % .....

r,,.

Interpretations of Heytmg's arithmetic

23

(b) Terms (I) Let t ~ h . . . . . t,,. t~ =- M(t~). Then we define:

/~({tl):~ Axe'i, ..... Ax'~,. (2)

M(WI"~U W~'~):-=

i

if ]V/(Itr} m A,

AxMU],xMj>~ ~

if gT'/({(rl~A.

U is defined with use of the functionals for simultaneous recursion by the following equations:

Ux~'O = 1f4( W~"~)x ~,

UxMNy = IQ( W~"~)x rc~.

{ AXM(IC4(W,)~.,[i,X ~ . . . . . j,,xrC']j,,+,X r;~)

,3,

( IVI

U .... •~-w

if/~/({(r})

~.,I.

w~;','):~

AxtC,(jCd(W, ) . . . . [j,x ~ . . . . . j , , x ~ . M ( W ) . I,+lx ~ .]l,,+~x ) otherwise.

w, and w2 are defined by

with /~(w")-=wi?

', . . . .

wT",

w~?,, ....

w~;,.

L e n u n a 4.10. (1) ~t((rl ~ ./¢:~/~7/(r')-= A for all terms r* of HA~,M.

(2) For any term t'" of HA~,~ it holds that IVl(t) is a term of HA~( ~ of type 1Vl(¢r) (/f/~/((r) ~ A). (3) lVl(t) =- t for all terms t of HA,,,(). (4) FV(~i(0) ~/~/(FVO)). (5) /(;/(r,[s]t --- ~/(r)M,.,[M(s)]. Definition 4.11. A mapping M of formulae of HArM to formulae of HA,,<>: ,t

(I)

M(r" = r ' ) : ~

if/~/((r) -~ A,

d,=r(,A

. . . Ar',{.... t;~', if lC4(r")=--r'(,. . . . . r'(,o. ff/l(t ¢" ) ~ t'(,. ....

(2) M ( A ~ B). M ( ~ x A ) are defined as usual. (3) Let w '~ be a sequence of variables. (a) Let /V/(w")=-.t. Then M(A w " ¢ W ~'*~A ) : m M ( A ) , (b) Let

1ft(w")

=- w i ? ' .

....

w'L;-,

w~i', . . . .

w~.~r,

t;:..

24

M. Stein

Let WI :~--- W11 . . . . .

~ln ;

W2 "*N W21 . . . . .

W2m,

Then we define

M ( A w" e W "~ A ) : ~ ~ Vwlx~M(A)~'~[lf4(W)x~] I. V w j M ( A )

if M~W)÷ A. otherwise.

For use in Section 5 we show that thc definition of the ~---translation of terms is very simple. We write & i" instead of &(tr), ¢qt) and get the following properties: =- o.

,T(r) =- e(-~).

/,r-] =- a .

If t is a basic term of linear type, then iv~: t. If i is a basic term of type (r with d ' ~ A . then t--=A. For basic terms, there remain two cases only: (1) R,~-- ~ if ~ - ~ ÷ ~ A . (2) w " - - w e if o ' ~ , ~ A . where w ~ may be chosen in such way that different w ° are m a p p e d to different -d2~r"

Furthermore we get

a(b)=-.a(f)L

Ax"a=-- ~ Ax'~a

if ,~ =-=-A. otherwise:

Its . . . . . t,,I=~WIUW2=- U U-~A. w~ ~M

This is the part of the definition we need for Section 5. If we look now at the x-interpretation of (A w e W ) A . we see that - - compared with the finite notion of set the A-translation of formulae g i v e s - - t h e x-translation yields the "biggest" notion of set:

(A w ~ W I A =--VwA. So all rules of HA,... concerning b o u n d e d qu~,ntification will become trivial. L e n u n a 4.12. ( 1 ) For any M-formula A. M ( A ) is a formula ,9[ T~M. or M ( A ) =- A. (2) F V ( M ( A ) ) c/V/(FV(A)). ~3) Let/5 be a function as in 2.2 such that P =/51 {A: A formula of HA.,< >} is a function as in 4.4. Then [br all formulae of HA°.~ it holds that

M ( A M~") ~ AMP,

IVJ(AMo) ~ Amp.

Theorem 4.13. (a)

Bt . . . . . B . , H A ~ . t A ~ or M ( A ) ~ A.

M(BO ..... M(B.)~----M(A) HA~

Interpretations o f H e y t i n g ' s arithmetic

(b)

Bi . . . . . B,,~

T, ~.1

A ~

M(BO . . . . . MtB.)~

"I"~M

25

M(A)

or M ( A ) ~ A. Proof. By induction on the length of the proof of A the reader ,'nay proceed as in the proof of Theorem 3.8. CoroUary. H A . ~ is conservative over HA,~ ). Theorem 4.14. (Soundness Theorem for MIP-interpretations of HA,o<>). Let P be a function as defined in 4.4 which satisfies the following conditions: (1t (2) (3)

A , . . . . . A,,~---HA.o,,A ::> P(AO . . . . . P(A,)~-HA~,,P(A). HA~>w-P(A}/',P(A--~B)--~P(B). P(A~[t])=-P(A}~[t].

Let there exist a fimction [~]br ~lA,o~ satisfying the conditions of 2.2 and 2.5. such that [~ I {A : A formula of H A ~ >}= P. Let A~ . . . . . A,,,

B:

BMv=-By V z B ,

A~I'=-3vi Vwifit,.

Then there are terms W, (or each w, ~ ,I and there is a sequence 9 of terms such that t P ( A , L £ VxMA,[v,. W~x]~----- F3[g, z] i-- I

i=l

HA~,,

and FV(W,)c_ FV{B)U U FV(A)U{v, . . . . . v.. z}. i-:l

FV(9)c_ FV(B)U 6 FV(A)U{v~ . . . . . v.}.

If MP is the M-interpretation. the deduction of B[ ~.. z] can be found in T^M. ProoL The proof is similar to that of Theorem 3.9 and left to the reader (use 2.5, 4.10. 4,12. and 4.13). Corollary 1. Soundness Theorems hold for lhe M- and Mq-interpretations (ME •l\{0}) and for the mr- and mq-interpretation. Corollary 2. HA<> is conservative over T^M for M e N \{0}. This holds because A m=- A for M-fimnulae A.

M. Stein

26

Theorem 4.15. (Characterization of M-interpretations). Let M~.~ be the scheme (Vw A - - , B ) - - ~ B W ( V x ~ A [ WxC'] ---~ B) ]'or n-formulae A, B of HA~< >, n ~ ' J . Let M~ be the prime formula (/= 0. Let M e {~-}UT'q\{0}. Then let IP M be the scheme (Vw A --~ 3x B) ~ 3 x ( V w A --" B) for M-formulae A of HA.,~ >, arbitrary formulae B of HA,,,~> and x~ FV(A). Let AC be the scheme V x 3 y A --, 3 Y V x A [ Yx ] for form ulae A of H A,.~ ~. Then the theorie~ HA,,,~ ) + {A ~'~AM}, and H A ~ + IPM + M~M, + AC are equivalent for M e ~J\{()}) U {~},

ProoL The proof corresponds to the proof of Theorem 3. Ill. The only difference is that Mr(M(M0 (Mu.~ in HA,,,,,.1) is not an instance of M~4~ (in HA,.< ~) but only equivalent to an instance of M (in H A ~ ~).

Remark. For K =-~7- the characterization theorem is the well-known characterization for the mr-interpretation.

5. Realizing terms of interpretations In this section we will consider only derivations in HA,,< >. So we can speak of mr-interpretation instead of 7z-interpretation. because both interpretations are identical in this case. If II ~s a derivation in HA,,,~ > of A. and A ~4 ~3vMVv.'MA~, then II M

A~d eM, w . l and the uniform proof of this f a c t - - f o r all ME~IU{:~} - l e a d s to the question whether we can compare the realizing terms gm of M-interpretations with the realizing term f ~ . ~ f , ~ of modified realizability. First. we see that type and structure of ~M, M e ~:q, are "'unnecessarily complicated" from the point of ~iew of mr, The part~; of ~ which are to~ complicated are characteriscd by having a non-linear type. In the example following Definition 4.11 wc developed a very simple method for eliminating these parts: the mapping er~.d', t,-* { maps c-types to linear types and terms of HA.,M to terms of HA,,,~>. The following theorem shows that only the unnecessary parts (from the point of view of m r ) ~ and ~lo other parts - - are cancelled.

Interptetutums of Heyting's arithmetic

27

Theorem 5.1. Let 11 be a derivation in HA~< > with :ndformula A Let

If I!u

A~,[r~,,. wM]

and

llmr Am~[rm~]

then ~7. = r . . .

Proof. T h e proof consists of two simple but rather long inductions: (I) Show by induction on the structure of B that. if B M m:ly M VZMBM we can choose y,,,, such that B'"'~3y,,,~Bm~. and ~ Ym~. (2) Show by induction on / / - - r e s p e c t i v e l y on the definition of 11M following T h e o r e m 2.5. that TM -= rm~. Part I of this proof supplies the start of this secor~d induction. If we consider M-interpretations and A-interpretation as "'independent" in'rpretations (this means, if they are not developed from the concept of M ~nterpretation). and if we want to get a corresponding result for these interpretations, we will first ha~e to m a r k - - f o l l o w i n g the proof of the soundness thcorems - - those parts of types and terms which are "too complicated" from the point of view of mr. Cancellation of marked parts then maps /:~, f~M to 6m~. (This method was outlined by Minc in [5].) We do not need to go through this procedure, because M - i n t e r p r e t a t i o n s mark the "complicated parts" (from the point of view of mrl of cM (A a formula of HA.,< >). using the type symbol {-}. and terms of t~on-linear type. The result of T h e o r e m 5. I can be applied to realizing terms of interpretations: all interpretations 1 we dealt with in the preceding sections have the property that they lead to a terr:l tm with ~--A[tm] if a closed deduction II of a formula I x A - - 3 x A not necessarily c l o s e d - - is given. In particular if II is a derivation in iqA,.,~ ~ and M ~ : h the realizing terms I,M are terms of HA~,M but will he of linear type and will contain free variables of linear type only. If we want to know whether different interpretations yield the same realizing terms, we h a v e - - w i t h regard to T h e o r e m 5. I - - t o deal with the following question: U n d e r what conception of e q u a l i t y - - i f at a l l - - h o l d s t = }'? Using very rough restrictions oq t. we get the result that t and t have the same normal form. Definition 5.2. An c-type is simple, if ~ ~ r

or ~i-= A.

Theorem 5.3. Let t be a term of HA.oM of linear type or. containing free variables of simple type only. Let t contain the recursor only if its type is linear. Then t and t hat'e tire same normal form.

2~

M. St.ein

Proof. By induction on the structure of the normal form t N of t. it is easy to show that t r~ is a term of HA.,( > For instance, if t N =- abt " " • b.. a ¢ a'(b.L the case that a is not of linear type can be excluded: variables of non-simple type are not allowed, a ¢ Axa because t t" is in normal form. and a ~ R only if the type of R is linear. So the induction hypothesis applies to the case a ~ R. in a second step we can show thai

t> t' => ~> ?

or

7-~ t-;

This leads to t->tN=-t ~ This theorem cannot be generaiised. If t :=--h u ° s d f "~ with s :'~ R(Ay"Wm~"U"'~ . 0)(AU m~ .ll)(u").

then t and [ are of different normal form. though extensionally equal, If we want to prove t = t- with less restrictions on the type of the recursor contained in t. we need stronger means for proving equations.

Definition 5.4. E - H A . . ) is the theory HA,,,< >+ t~- and Ti-equahty: (~)

t=t

'

~xt = AxI'

(rD A).(tx)= t,

if x ¢ FV(t).

o~E-I~IA,,,~ ) is E-HA,,. )+ the foUowing ~.-rule:

r"[0] = r'[0]

r"[k]= t"[k]

r " f s ] = C'[s] if s is a term of type o and all v k only contain equations. W e write E , - - A instead of E - H A , , ( ~ - - A . toE~--- A instead of (oE-HA=< r~.-A.

Definition 5.5. The predicates l.in ("Linearisable") and Lin' ("lJncarisable under substi,'ution") are defined as follows: Lin is defined for all terms t of HA.,M which contain free variables of simple type only. by induction on the type of t: (1) Lin(t"J for all terms t~ with (r=-A, (21 Lin(t'~):C~E~.-t = i, (3) L i n ( t ' ~ ' ) : c : ~ L i n ( t ( b " ) ) for all b" with Lin(b), Lin'(t):¢~Lin(tx[b])

for all sequences x such that x contains all free variables of t which have a non-simple type; x may contain some other free variables of t for all sequences b of appropriate type with Lin(b).

L~mma 5.6. For all linear types cr the variable x " is linearisable u n d e r substitution, a n d L i n ( b " ) ~ E F b " = b",

Interpretations of Heytmg's aridm}etic

P r o o f . T h e l e m m a is trivial for t r y 0 . N o w let c r y ( o , ) . p r o v e d for all o, (a) L i n ' ( x " ) m e a n s (I) E ~ - - x " b t "" " b . = x " b , " ' " b . ; (2) E ~ t~6, . . . b,, = r'h~...b~i if Lin(bi) a n d Lin(t), a n d bi of type o i. (1) holds by induction hypothesis, because x"b,

. •

• b, = - x b. ,.

b-.

. . .

=-

x

2q

• • (o, j0. ',he l e m m a be

" -b ~ " " • b,,.

{2) holds by definition of L i n ( r ' ) . t~ (b) L i n ( b " ) implies E F b " x = -b "- x , x = - - x ,u, , . . . . x,o because Lin(x,) by induction hypothesis. So E ~ - - - b = ~ x ( b x ~ • • • x , , ) = ;~x(bx, • . . x,,)~ X~(/~:~, • • • ~.)=/~.

Theorem 5.'/. L e t t be o f l i n e a r type. c o n t a i n i n g I:ariables o f s i m p l e type only. T h e n (I) E ~

t = ~ i f t c o n t a i n s the recursor R,,~)~,,~ o n l y i f ": is linear, or • =- (crlu with

u l i n e a r a n d 6 - ~ .I.

(21 oJE~---- t = i-. P r o o t . a d I : Because of L e m m a 5.b it is sufficient to prove Lin'(t) for all terms of HA.,~,. T h i s is d o n e by induction on t h e structure of t. T h e only interesting case is t ~ R. of type (orr)'ror with r-= (cr)v. o linear and 6" ~-~I . W e will show by an application of the rule lind) that (for W of type tr) E~--Rabs = A W . RObs

for arbitrary linearisable t e r m s a. b. s. This leads to Lin'fR). because ( W a t e r m of type ~r)

E,---Rabs~/e=Rdbg~

by L e m m a 5.(4. if c-=c~,. . . . . c,~-with L i n ( q ) . o ~ ( o l ) . • - (o,,Io. T h e induction runs as follows: E~---- R a b O = b = A W " . Rab(NyJ(W)=

b(W) = AW.

ay(Raby)(W) Rab(Nyl(W)

b(W) = hWb=

[Raby = AW. = ay(hW.

RiibO.

Rdby]

Raby)(W)

Rab(Ny)(W)

AW.

ay(XW.

RaEy)(W~-~ay(Raf~y~

= fiy(/~t~/~y) = i~fi/~(Ny)

AW. RaMNy} (W) =AW. R~b(Ny) (,I) Rab(Nyl= R a b y = A W . ff~db},, --~ R a b ( N y )

A W. R~b(Ny)

= AW./~ti/~(Ny)

A n application of lind) now leads to (1). F o r the e q u a t i o n * we use that a is linearisable by hypothesis, y is trivially linearisable, a n d A W . ~ a E y is linearisable because /~d/Lv is a t e r m of HAm<7.

30

M. Stein

F u r t h e r m o r e , I~/';--- 'r~ " and so h W . / , ~ / ~ y ~ Rg~by. ad2: W e have to modify the predicate Lin such that Lin(t'):C~coE~-- t = t. T h e rest of Definition 5.5 r e m a i n s the same, and L e m m a 5,6 still holds. Again. we are only interested in the case a ~ R. Let R be of Type (oft)to'r. "r :~ (~r}o, rr ~ ~r~ • ' •. ~r., Let a, h. s be linearisabl¢ t e r m s of a p p r o p r i a t e type. W e snow by induction on k that for all n u m e r a l s k the following holds: for all linearisable c of type ~r t h e r e exists a derivation _Vk(c) Rabkc = R~l~kE ~ . ( c ) is the derivation for the following chain of equations: RabOc = hc = bE =/~/~1}(, -~k ~d c ) is the following derivation:

Rah(k + Ilc = a k ( R a b k ) c

II a k ( R a b k ) c = ik(RatTkle

T h e derivation II exists b e c a u s e Rabk is linearisable by induction hypothesis. By an application of the oJ-rule we get the following derivation: -v.(c)

.....

'q(c)

....

using that s is linearisable and L e m m a 5.6. So L i n q R l holds. The p r o o f remains the s a m e if we consider R as the functionals for simultaneous recursion. T h e o r e m 5.8. l,et I1 be ~t closed derivation in HA,,,~ , Of 3X A, ~x m tlol necess,wih' closed. Then ,oE b--- Jr I11~.. (,,1~ .... t~n - t,., for oil MEI i\{0}. =

(For reason ot simplicity we have written t,. tm~. tm instead of t . . . in,,., t.~.) P r o o L T h e terms t,. tm are terms of HA,,,>. T h e c o r r e s p o n d i n g t e r m s t,,. tt~ - - which we get by M - i n t e r p r e t u t i o n s - - ~ may hc terms of HA,°st. but Ihey ;.Ire of linear type. and they contain variables of linear type only. So we gel by ' l ' h c o r c m s 5.1 and 5.7:

This implies toE~-- lM = tmr.

Interpretations o( tteyting's arithmetic

31

By T h e o r e m s 3.8 resp. 4.13 (which are not spt~iled by o u r t0-rule ~lnd the additional rules for the A - o p e r a t o r ) we get: oJE~---~ (t,, : tm~t. this m e a n s , J E ~ - 7, (ta) = t,,,r ttl ~,, = tmr

toE~-- M(tsl : t,,,,), this m e a n s

t M = tmr. T h e results of this section - - especially those of T h e o r e m s 5.7 - - were o b t a i n e d by analysing the structure of terms. T h e "'weakness of set-symbols'" in our system was of g o o d a d x a n t a g e to us. H o w e v e r . if we want to get f u r t h e r results, we have to d e v e l o p difterent m e t h o d s : in a n o t h e r paper. [7]. we will prove that for clo.sed ] x A all terms t~ have the s a m e normal form ( M E ~ i L) {:~}1. This can not be d o n e by the m e t h o d s of this secti~,n, b e c a u s e for closed terms of linear type of HAoM. t d o e s not always have the s a m e normal form as t-. as the e x a m p l e following T h e o r e m 5.3 shows. T o get the result on closed existence formulae, we will need an explicit analysis of the process of normalisation in c o r r e s p o n d e n c e with interpretation. This analysis will show that t~f>t~,. M~[~t_){~}. t~ being the term of H A ~ with ~--A[t,~ ] which we get by normalising the derivation of 3x A. (This implies t. 7" t~.,. t,~,7"t~.L ira particular, we see that t~, n e v e r can have the s h a p e as the term following T h e o r e m 5.3. and if it contains such term as a subterm, this s u b t e r m will bc eliminated by s o m e reduction step of t~.

References [ I I J Dillcr and W. NahnL Einc Variante zur Dialectica-- Interpretation der Heyting-Arithmetik cndlicher l),pen. Archly, tiir mathcmatische Logik und Grundlagenforschung 16. ~1t~741 4~-(~tL 12] J. Diller. Functional interpretations of Heyting's arithmetic in all finite types. MC Tract l(llL Proeeedings Bicentennial Congress Wiskundig Genootschap, part 1, 1'~7tL 14')-I 76 [3] K. G,3del. Uber eine hisher noch nicht beniitzte Erweiterung des finite=~ Standpunktes. Dialectica 12 q lt,~SSI. 2S~k-2sT. 14I G. Krciscl Interpretation of analysis by means of constructive tunctionals of finite type. in: A. Ite~.'ling cd,: (',t~nstructi~,ity in Mathematics (North-Holland, Amsterdam. lt/5~)) 111t-12S. t5t ( i . E Mine, Sla~ifit~. of E-theorems and program ~erification. in: Semiotika i informalika. N 12. Mo.~cov,, VINI'I1/I It)7tl. 73-77 [in Russian], I¢,1 M. Stein. lnterpretationen der Heyting-Arithmetik endlicher Typen. Archiv f~ir mathematische L,3gik und Grundlagenforschung 19 (lt)7b;) 175-189. [71 M. Stein. A general theorem on existence theorems, to appear in ZML. [~] A,S~ Troelstra. Metamathematical investigation of intuitionistic arithmetic and analysis. Lecture Notes (Springer, Berlin. 1973)~